The process of joint

diagonalisation is repeated until convergence.

For the right-hand side is a member of a language further down the hierarchy since @ is a term of [L.sub.[Gamma]] as are the canonical representations of ordinal addition, the resources for diagonalisation and so forth.

(39) Or of that hierarchy of G6delian proof predicates characterising ever stronger notions of proof which is generated by adding as an axiom at each stage [Alpha] the hitherto undecidable diagonalisation on "unprovably", [Beta] [is less than] [Alpha].

These results have been shown to emerge [47,109,110,112,176,177] from SO(10) with gauge fermion singlet extensions by following the explicit block diagonalisation procedure in two steps while safeguarding the hierarchy [M.sub.N] [much greater than] M > [M.sub.D], [[mu].sub.S] with the supplementary condition [[mu].sub.S][M.sub.N] < [M.sup.2].

Whereas the heaviest RH neutrino mass matrix [M.sub.N] separates out trivially, the other two 3 x 3 mass matrices [M.sub.v], and [M.sub.S] are extracted through various steps of block diagonalisation. The details of various steps are given in [112,176,177].

If t is any term, let its

diagonalisation be the term obtained by substituting <t> for each free variable in t.

More fundamentally, paradoxes in Bii use diagonalisation, or some similar mechanism, to establish something of the form [alpha] [equivalence] [logical not][alpha], and hence [alpha] and [logical not][alpha]; whereas paradoxes in Bi give independent arguments for each of [alpha] and [logical not][alpha].

In this, [phi](x) is "x is a definable real number between 0 and 1", so that w is the set of such reals, DR; [psi](x) is "x is definable" and [delta](x) is diag(x), a real, defined by diagonalisation on x, in such a way as to ensure that [delta](x)[not an element of!x.